Стоит мне открыть арбуз, как дети сразу приходят на кухню.

This is a set Russian construction meaning as soon as I... , ... happens or the moment I... , ....

So:

Стоит мне открыть арбуз, как дети сразу приходят на кухню.

means roughly:

As soon as I open a watermelon, the children immediately come into the kitchen.

Literally, стоит normally means it costs or it is worth, but in this pattern it does not mean that in a direct, literal way. The whole expression functions like a trigger:

• Стоит мне... = it only takes me to... / as soon as I...
• как... = and then / when immediately...

It often suggests that the second thing happens very quickly and almost automatically.

Because this construction uses the dative case for the person involved:

• мне = to me / for me
• not я = I

So Russian says something like:

• Стоит мне открыть арбуз...
• literally: It only takes for me to open the watermelon...

Other examples:

• Стоит тебе позвонить, как он отвечает.
  As soon as you call, he answers.
• Стоит ей улыбнуться, как все радуются.
  As soon as she smiles, everyone becomes happy.

This dative after стоит is just part of the pattern and must be learned as such.

Because after стоит мне Russian uses an infinitive:

• Стоит мне открыть...
• Стоит ему сказать...
• Стоит нам уйти...

So the structure is:

стоит + dative person + infinitive

This is different from English, where we would normally use a finite verb:

• As soon as I open...
• not It is worth to me open...

In Russian, the infinitive is the normal form in this construction.

Открыть is the perfective verb, and here that makes sense because the sentence refers to a single completed action that triggers another action:

• открыть = to open / to open up completely
• открывать = to be opening / to open repeatedly / to open in general

In Стоит мне открыть арбуз..., the idea is:

the moment I finish opening the watermelon, the children appear

Perfective is very common in this pattern because it highlights the action as a completed event that causes the next one.

If you used открывать, it would sound different and usually less natural here.

Yes, как often means how, but in this construction it has a different function.

In Стоит мне открыть арбуз, как дети сразу приходят на кухню, как introduces the result that follows immediately after the first action.

So here it is closer to:

• and then
• when immediately
• as soon as

You should learn стоит..., как... as a whole paired pattern.

A similar feeling in English is:

• No sooner do I open the watermelon than the kids come into the kitchen.

So как here is not asking how; it is linking the two parts of the sentence.

The present tense here expresses a habitual or repeated situation.

The sentence means something like:

Whenever I open a watermelon, the kids immediately come into the kitchen.

So Russian uses present tense because this is not about one specific past event. It describes what typically happens.

• Стоит мне открыть арбуз, как дети сразу приходят на кухню.
  As soon as I open a watermelon, the children immediately come into the kitchen.

If you wanted a one-time past event, you would use past forms, for example:

• Стоило мне открыть арбуз, как дети сразу пришли на кухню.
  As soon as I opened the watermelon, the children immediately came into the kitchen.

Because арбуз is the direct object of открыть, so it is in the accusative case.

For an inanimate masculine noun like арбуз, the accusative singular is the same as the nominative singular:

• nominative: арбуз
• accusative: арбуз

So the form does change grammatically, but it just happens to look the same.

Compare with a feminine noun:

• открыть дверь
  Here you can see the accusative form clearly.

Because with the verb приходить when meaning to come to a place, Russian commonly uses на with certain destinations, including кухня:

• приходить на кухню = to come into/to the kitchen

This is just how Russian usually expresses movement to some locations. You often have to learn the preposition together with the noun or verb.

Compare:

• идти в школу = to go to school
• идти на кухню = to go to the kitchen
• идти на работу = to go to work

So на кухню is the natural choice here.

Сразу means right away, immediately, or at once.

Without it, the sentence already means that the second action follows quickly because of the стоит..., как... structure. But сразу makes that immediacy even more explicit.

So:

• как дети приходят на кухню = and the children come into the kitchen
• как дети сразу приходят на кухню = and the children come into the kitchen immediately

It adds emphasis: the kids waste no time.

In Russian, bare singular nouns can often have a general meaning depending on context.

So открыть арбуз here can mean:

• open a watermelon
• or, depending on the situation, open the watermelon

Because the rest of the sentence is habitual, English will often translate it more naturally as a watermelon:

As soon as I open a watermelon, the children immediately come into the kitchen.

The Russian sentence itself does not need an article, since Russian has no articles like a or the.

Yes. That is actually a very good stylistic match.

Possible translations include:

• As soon as I open a watermelon, the children immediately come into the kitchen.
• The moment I open a watermelon, the children immediately come into the kitchen.
• No sooner do I open a watermelon than the children come into the kitchen.

The last one matches the Russian structure especially well because it has the same sense of one action instantly triggering another.

To some extent, yes, but the basic pattern is usually kept clear:

• Стоит мне открыть арбуз, как дети сразу приходят на кухню.

This is the most straightforward order.

Russian word order is flexible, so you might move things around for emphasis, but beginners should keep the standard pattern:

Стоит + dative + infinitive, как + clause

For example, moving сразу is possible:

• Стоит мне открыть арбуз, как дети приходят сразу на кухню.

But that sounds less natural than the original. So the original word order is a good model to copy.

The past-tense version uses стоило:

• Стоило мне открыть арбуз, как дети сразу пришли на кухню.

This means:

As soon as I opened the watermelon, the children immediately came into the kitchen.

Compare:

• Стоит мне открыть... как... приходят...
  habitual / what usually happens
• Стоило мне открыть... как... пришли...
  one past event

This is a very useful pair to remember.